H1011 Quizzes

Quiz 2: Sinusoidal functions and proportionality
Question 1 Questions
Which of the following correctly describes the function shown ?
 a) $y=2cos\frac{4}{3}\left(x-\frac{3\pi }{8}\right)$ b) $y=2sin\frac{4}{3}x$ c) A sinusoidal function with amplitude 2, period $\frac{3\pi }{2}$ and mean level 0. d) All of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Find the amplitude and the mean level of the function $f\left(x\right)=2sin\frac{2\pi }{3}\left(x-4\right)+1$.
 a) Amplitude = 1, mean value = 2. b) Amplitude = 2, mean value = 1. c) Amplitude = 3, mean value = 4. d) Amplitude = 4, mean value = 3.

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Find the period of the function $f\left(x\right)=2sin\frac{2\pi }{3}\left(x-4\right)+1$.
 a) $\frac{2\pi }{3}$ b) $\frac{3}{2\pi }$ c) $\frac{1}{3}$ d) $3$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Period = $2\pi ∕\frac{2\pi }{3}=3$.
Find the period and the amplitude of the function $f\left(x\right)=4sin\left(3x-5\right)+\pi$.
 a) Amplitude = 4, period = $2\pi$. b) Amplitude = 5, period = $2\pi$. c) Amplitude = 4, period = $\frac{2\pi }{3}$. d) Amplitude = 5, period = $\frac{2\pi }{3}$.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
If $y=4sin\left(3x-\pi \right)+5$, which of the following is true ?
 a) $-4\le y\le 4$ b) $1\le y\le 9$ c) $1\le y\le 5$ d) $-4\le y\le 9$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Which of the following has period $\frac{\pi }{2}$ and amplitude 4 ?
 a) $y=4sin4x$ b) $y=4cos\frac{\pi }{2}x$ c) $y=\frac{\pi }{2}sin4x$ d) $y=4cos4x$

Choice (a) is correct!
Note that 4 is also correct.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Note that 1 is also correct.
Which of the following has period 2 and amplitude 3 ?
 a) $y=2sin3x$ b) $y=3cos2x$ c) $y=2sin\pi x$ d) $y=3cos\pi x$

Choice (a) is incorrect
1 has amplitude 2, period $\frac{2\pi }{3}$.
Choice (b) is incorrect
2 has amplitude 3, period $\pi$.
Choice (c) is incorrect
3 has amplitude 2, period 2.
Choice (d) is correct!
Which of the following has period $4\pi$ and amplitude $\frac{1}{2}$ and is such that $f\left(0\right)=\frac{1}{2}$ ?
 a) $y=\frac{1}{2}sin4x$ b) $y=\frac{1}{2}cos\frac{1}{2}x$ c) $y=\frac{1}{2}sin\frac{1}{2}x$ d) $y=\frac{1}{2}cos4x$

Choice (a) is incorrect
1 has amplitude $\frac{1}{2}$, period $\frac{\pi }{2}$.
Choice (b) is correct!
Choice (c) is incorrect
3 has amplitude $\frac{1}{2}$, period $4\pi$, but $f\left(0\right)=0$.
Choice (d) is incorrect
4 has period $\frac{1}{2}$, period $\frac{\pi }{2}$.
Expand $sin2\left(x+\frac{\pi }{3}\right)$.
 a) $\frac{\sqrt{3}}{2}cos2x+\frac{1}{2}sin2x$ b) $\frac{\sqrt{3}}{2}cos2x-\frac{1}{2}sin2x$ c) $\frac{\sqrt{3}}{2}sin2x+\frac{1}{2}cos2x$ d) $\frac{\sqrt{3}}{2}sin2x-\frac{1}{2}cos2x$

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{rcll}sin2\left(x+\frac{\pi }{3}\right)& =& sin\left(2x+\frac{2\pi }{3}\right)& \text{}\\ & =& sin2xcos\frac{2\pi }{3}+cos2xsin\frac{2\pi }{3}& \text{}\\ & =& -\frac{1}{2}sin2x+\frac{\sqrt{3}}{2}cos2x& \text{}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
The sinusoidal function $f$ is illustrated below.
The function $g$ has mean level 0, amplitude half that of $f$, and period twice that of $f$. Find the equation for the function $g$.
 a) $g\left(x\right)=\frac{3}{2}cos2x$ b) $g\left(x\right)=3cos\pi x$ c) $g\left(x\right)=\frac{3}{2}cos\pi x$ d) $g\left(x\right)=3cos2x$

Choice (a) is correct!
Since $f$ has period $\frac{\pi }{2}$ and amplitude 3, $g$ must have amplitude $\frac{3}{2}$ and period $\pi$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect